Saturday, May 28, 2016

Hi everyone— before we start, just wanted to remind you, the Math Mutation book is out! To order, you can follow the link at mathmutation.com or just search for it on Amazon. And if you ever pass through the Portland, Oregon metro area, I’ll be happy to autograph your copy. If you like it, posting a positive review on Amazon would be really helpful. Now on to today’s topic.

As you may recall, one of the topics we covered in some previous podcasts, and in the Math Mutation book, is the idea of “Cognitive Biases”. These are well-known ways in which the human brain has a natural instinct to think in ways that violate basic laws of logic and mathematics. One classic example is the Anchoring bias: if asked a question that has a quantitative answer, you will tend to give an estimate close to numbers you recently heard. For example, suppose I arrange separate discussions with two people to estimate how many listeners Math Mutation has. With the first one, I start by asking “Does Math Mutation have more than 100 listeners, or fewer than 100?”. But with the second one, I open with “Does Math Mutation have more than 1 million listeners, or fewer than 1 million?” If I then ask both of them to estimate the total number of listeners, the first will probably come up with a much smaller estimate than the second, even though neither has any objective information to justify a particular number.

After reading the chapter in my book, my old Princeton classmate Tim Chow pointed out that calling this a “Cognitive Bias” might not be justified. Sure, the listener technically has no information to support the larger number in the second case— but in cases where we are talking to another human being, we trust them to provide relevant information. This includes both direct statements of facts, and implications that might not be directly stated. If I ask you whether Math Mutation has more or fewer than 100 listeners, I am implicitly communicating the information that the 100 number is pretty close, even though I have not rigorously declared this to be a relevant fact. So if this number isn’t close, I have essentially misled you with false information— the fact that you trusted me and used the wrong number is my fault, not some flaw in your mental logic. Thus, this “Cognitive Bias” is really a social manipulation.

Now, if you’re familiar with the literature on this topic, you might point out an interesting experiment that seems to refute this. In this experiment, subjects saw a roulette wheel spin, then were asked the percentage of the United Nations countries that were in Africa. Even though there is no logical reason for them to suspect the roulette wheel had advanced knowledge of geopolitics, their answers were still biased towards the results they saw on the wheel. Many similar experiments have been carried out. Doesn’t this provide irrefutable proof that this really is a cognitive bias?

Not so fast. This is a very artificial situation. Maybe when asked to guess a number about which they have absolutely no idea, they just grab any arbitrary number they can think of, which will tend to be one they saw recently. They aren’t following some flawed cognitive process, they just don’t have any reason to pick any particular number. Again, this doesn’t really indicate a mathematical flaw in their reasoning— they don’t think the number they picked has a particular logical justification. Not knowing an answer, they just defaulted to what was at the top of their head.

Most of the other well-established Cognitive Biases are open to similar criticisms. Another example is the Conjunction Fallacy: suppose I tell you that Joe is a Princeton mathematics graduate and chess champion, and then ask you to choose the more likely of two statements. 1. “Joe is now a physics professor.” 2. “Joe is now a physics professor and head of the local Math Mutation fan club.” You will likely choose option #2, since it seems like this kind of guy should be a Math Mutation fan. But on reflection, option 2 must be strictly less likely than option 1, as it takes the same basic fact and adds an additional, more restrictive, condition. But again, there is information being communicated between the lines: if I give you those two choices, you probably interpret #1 as implicitly stating that Joe is NOT president of the Math Mutation fan club. I didn’t say that, but the additional choice in the second option made this a very reasonable inference. Once again, it can be seen as more of a social manipulation, where I leveraged typical communication conventions to imply something without actually stating it, and the implication is not strictly justified by mathematical logic.

We should point out, though, that even if the so-called “Cognitive Biases” are not truly flaws in the logic of the human brain, they are still important psychological effects to be aware of for many reasons. For example, let’s take a look at some practical applications of the Anchoring bias. It’s well known that when negotiating prices in business, your opening offer can set an anchor that affects the entire discussion. Negotiating business contacts usually have some level of trust in each other, and taking advantage of this to establish a good anchor is a smart, though slightly manipulative, technique. On another note, suppose you’re a surveyor trying to get accurate estimates in a survey or questioning experts on a difficult topic. You need to be careful not to include some kind of number in the question that might unintentionally influence the result. So knowing about Anchoring is still very useful, whether you call it a true cognitive bias or a simple persuasion technique. In general, I still believe the Cognitive Biases are worth studying and raising awareness of, though maybe more as social or linguistic phenomena than as true flaws in the human mind.

Sunday, May 1, 2016

I was sad to hear of the recent passing of Umberto Eco, one of my favorite contemporary novelists. Due to his long career as a professor of “semiotics”, his novels drew on a wide variety of historical, mathematical, and scientific ideas from throughout the past millennium. One that I read relatively recently was “The Island of the Day Before”, his 1994 story of a soldier marooned during the 1600s, the historical period during which the governments of Europe were desperately searching for a reliable way to measure longitude. You may recall our discussion of this multi-century quest back in episode 108. The key plot twist is that the main character, an Italian soldier named Roberto Della Griva, gets marooned on a ship that is trapped within sight of the International Dateline, and an island which sits beyond it.

Della Griva attaches a mystical significance to this line, as is implied by the book’s title. He somehow convinces himself that if he could just cross it, it would mean he was traveling back in time. If he will manage to swim across the line tomorrow, would it mean that today he should see himself swimming in the distance? Here is one amusing passage from the book:

“Indeed, as he sees it distant not only in space but also (backwards) in time, from this moment on, whenever he mentions that distance, Roberto seems to confuse space and time, and he writes, "The bay, alas, is too yesterday," and ,"How much sea separates me from the day barely ended," and even, "Threatening rainclouds are coming from the Island, whereas today it is already clear . . . .”

While these speculations are rather absurd, this discussion got me curious about the actual history of the International Dateline. It has been recognized since ancient times that the time of day is slightly different as you travel to the east or west, but the idea that you might travel all the way around the world and gain or lose a day was only really conceivable in relatively recent eras of human history. The real history of the Dateline can probably be properly said to have started with Magellan’s circumnavigation of the globe. When the handful of survivors of that three-year voyage arrived home in 1522, they were surprised to discover that despite careful logging of their travels, they had lost a day. Here is a description from one of them:

On Wednesday, the ninth of July [1522], we arrived at one these islands named Santiago, where we immediately sent the boat ashore to obtain provisions. [...] And we charged our men in the boat that, when they were ashore, they should ask what day it was. They were answered that to the Portuguese it was Thursday, at which they were much amazed, for to us it was Wednesday, and we knew not how we had fallen into error. For every day I, being always in health, had written down each day without any intermission. But, as we were told since, there had been no mistake, for we had always made our voyage westward and had returned to the same place of departure as the sun, wherefore the long voyage had brought the gain of twenty-four hours, as is clearly seen.

As you can see, even though they were caught by surprise at first, the sailors were able to quickly realize their fallacy. Because they were traveling in the same direction as the sun, they had experienced one less day, but each day they had experienced was slightly longer. So there was no actual time travel, just an accounting error. A similar phenomenon was observed later by other circumnavigators, such as English explorer Francis Drake. This incident also inspired the famous surprise ending of Jules Verne’s “Around the World in 80 Days”. Even though the reasons for the gain or loss of a day upon circumnavigation were well known, I suppose it is vaguely possible than an uneducated sailor like Eco’s character could have attached a more mystical significance to the effect.

But even these odd experiences of sailors were not really common enough to motivate standardization of time zones and an international dateline, until the era of trains came along in the 19th century. Suddenly it was possible to move quickly and continuously between areas with different local times. Finally in October 1884, representatives from 25 nations met at an international conference in Washington, DC, and came up with the system of time zones we know today, based on longitudinal lines starting from the Greenwich meridian, and times derived by adding or subtracting from the Greenwich Mean Time, or GMT. To increase the chances of universal adoption, it was agreed that local islands and nations can move the timelines for convenience, which is why we see those squiggly time zone boundaries today instead of simple longitudinal lines. Amusingly, the French seemed to be insulted by the idea of a location in England defining the time zones: until 1911, instead of referring to Greenwich Mean Time, they referred to Paris Mean Time minus nine minutes and 21 seconds, which was equivalent.

Unfortunately, since these meridians and zones are all artificial labels, I’m afraid Della Griva’s dream of somehow using them for time travel would never quite pan out. Some modern writers ponder this idea as well, but I think they are mostly tongue-in-cheek. For example, travel author Bill Bryson has written:

“I left Los Angeles on January 3 and arrived in Sydney fourteen hours later on January 5. For me there was no January 4. None at all. Where it went exactly I couldn’t tell you. All I know is that for one twenty-four-hour period in the history of earth, it appears I had no being. I find it a little uncanny, to say the least. I mean to say, if you were browsing through your ticket folder and you saw a notice that said, ‘Passengers are advised that on some crossings twenty-four-hour loss of existence may occur’…, you would probably get up and make inquiries, grab a sleeve, and say, ‘Excuse me.’”

But somehow I don’t think Bryson really believes he lost a day of his life due to the shifting of a few time labels. I would be more concerned about the portion of my existence that is wasted while crammed into a tiny airline seat for half a day.

In a more serious vein, there was a solar eclipse last month that started on March 9 and ended on March 8, but it was actually traveling forward in time the whole way, although it achieved its peculiar timeline by crossing the International Dateline as it travelled. Also, you shouldn’t completely lose hope— one form of time travel across the dateline is possible. Remember that under the theory of relativity, if you travel by airplane at a high speed, you really do lose a tiny fraction of a second, as time slows down for you in relation to your friends on the ground. However, that works across any line, not just one artificial one.

Note that this podcast is intended mainly for audio consumption, so you will not see the numerous illustrations & diagrams you would find at most math sites, though these are linked in the show notes whenever possible.

The Math Mutation Book

The Math Mutation book, "Math Mutation Classics: Exploring Interesting, Fun, and Weird Corners of Mathematics", is now available. You can order it from Amazon at this link.

By the way, if you like it, don't forget to post a review on Amazon. And of course, if you're in the Portland, Oregon metro area, I'll be happy to autograph your copy sometime.

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